Introduction to Categorical Data Analysis

324 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
have you known personally that were victims of homicide?” It used Poisson
and negative binomial GLMs for count data. Here is a possible GLMM: For
response yi for subject i of race xi (1 = black, 0 = white),
log[E(Yi)] =ui + α + βxi
where conditional on ui, yi has a Poisson distribution, and where {ui} are independent
N(0,σ). Like the negative binomial GLM, unconditionally (when
σ>0) this model can allow more variability than the Poisson GLM.
a. The Poisson GLMM has ˆα =−3.69 and ˆβ = 1.90, with ˆσ = 1.6. Show
that, for subjects at the mean of the random effects distribution, the
estimated expected responses are 0.167 for blacks and 0.025 for whites.
b. Interpret ˆβ.
10.23 A crossover study compares two drugs on a binary response variable. The
study classifies subjects by age as under 60 or over 60. In a GLMM, these
two age groups have the same conditional effect comparing the drugs, but
the older group has a much larger variance component for its random effects.
For the corresponding marginal model, explain why the drug effect for the
older group will be smaller than that for the younger group.
10.24 True, or false? In a logistic regression model containing a random effect as
a way of modeling within-subject correlation in repeated measures studies,
the greater the estimate ˆσ for the random effects distribution, the greater the
heterogeneity of the subjects, and the larger in absolute value the estimated
effects tend to be compared with the marginal model approach (with effects
averaged over subjects, rather than conditional on subjects).